Well, the Nobel quote is wrong whichever way you turn it (for the reason outlined in the Wikipedia quote). But I wouldn’t be surprised that, even taking the average, there’s an uncertainty of a factor 10 about the correct number. Because all the factors used to determine the number are crude estimates indeed.
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Konrad RudolphAug 26 '12 at 19:50

2 Answers
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** Found the snippet that contained the reference. I think it's really worth including in its entirety.

A crude estimate of the amount of DNA within currently living
organisms can be made by noting that the length spanned by one base of
DNA is ~0.3x10^-12 km (Cook 2001).

The number of viral particles in the open oceans is ~10^30 (Suttle
2005). Assuming that there are twice as many viruses on land and in
fresh water does not change the global estimate very much at the
order-of-magnitude level. Thus, assuming an average viral genome size
of 10^4 bp, the total length of viral DNA if all chromosomes were
linearized and placed end to end is ~10^22 km.

The estimated global number of prokaryotic cells is ~10^30 (Whitman et
al. 1998), and assuming an average prokaryotic genome size of 3x10^6
bp yields an estimated total DNA length of 10^24 km.

With a total population size of 6x10^9 individuals, 10^13 cells per
individual (Baserga 1985), and a diploid genome size of 6x10^9 bp, the
amount of DNA occupied by the human population is ~10^20 km. Assuming
there are ~10^7 species of eukaryotes on Earth (~6 times the number
that have actually been identified), that the average eukaryotic
genome size is ~1% of humans, and that all species occupy
approximately the same amount of total biomass, total eukaryotic DNA
is ~10^5 times that for humans, or ~10^25 km.

Given the very approximate nature of these calculations, any one of
these estimates could be off by one or two orders of magnitude, but it
is difficult to escape the conclusion that the total amount of DNA in
living organisms is on the order of 10^25 km, which is equivalent to a
distance of 10^12 light years, or 10 times the diameter of the known
universe.

Bianconi et al. 2013 give an estimated lower bound of 3.72 × 10^13 (which, by the way, is approximately the geometric mean of 10^13 and 10^14).

However, from the table in their Supplemental Information (where estimates for about fifty different types of cells are added up), it is clear that the vast majority of these are the erythrocytes, also known as red blood cells: estimated at 2.63 × 10^13. While the second largest class are the glial cells (from the nervous system), estimated at 0.30 × 10^13. Three other classes (endothelial cells {vessels}, dermal fibroblasts {skin}, and platelets {blood}) add up to another 0.58 × 10^13. These, in total, is about 3.5 × 10^13, and we can stop there, because the uncertainty in all these estimates swamps all the other cell type populations, which are much smaller.

This means that if we only consider the 5 most abundant cell types in the body, we already get a very good estimate. Is it likely that we have missed a cell type of which there are a trillion cells in the body? It's not impossible, but unlikely, because it would be hard to miss all these cells.

Maybe there is a long tail (i.e. a large number of minor cell types, each with a small, but significant population) that wasn't considered? Together, these could add up to a significant adjustment to the above figure. I don't know.

This estimate is for a typical human person: 30 year old young adult, 70 kg, 1.72 m tall and with a surface area of 1.85 m^2. For this person, the figure is given with roughly a +-20% uncertainty.

Given all the above, it's probably safe to say "30 to 50 trillion".

(This obviously doesn't take into account any of the bacteria mentioned above.)